We imagine shining light on a lump of cold dense quark matter, in the CFL
phase and therefore a transparent insulator. We calculate the angles of
reflection and refraction, and the intensity of the reflected and refracted
light. Although the only potentially observable context for this phenomenon
(reflection of light from and refraction of light through an illuminated quark
star) is unlikely to be realized, our calculation casts new light on the old
idea that confinement makes the QCD vacuum behave as if filled with a
condensate of color-magnetic monopoles.Comment: 4 pages, 1 figur

A. Leibovich

B. Barrois

C. Alcock

C. Manuel

Cristina Manuel

D. Bailin

D. F. Litim

D. H. Rischke

D. T. Son

E. Fahri

E. Witten

G. 't Hooft

J. D. Jackson

K. Rajagopal

Krishna Rajagopal

M. Alford

P. Haensel

R. Casalbuoni

R. Rapp

S. Mandelstam

T. Schäfer

English

arXiv

Crossref

Illuminating Dense Quark Matter

We imagine shining light on a lump of cold dense quark matter, in the CFL phase and therefore a transparent insulator. We calculate the angles of reflection and refraction, and the intensity of the reflected and refracted light. Although the only potentially observable context for this phenomenon (reflection of light from and refraction of light through an illuminated quark star) is unlikely to be realized, our calculation casts new light on the old idea that confinement makes the QCD vacuum behave as if filled with a condensate of color-magnetic monopoles.We imagine shining light on a lump of cold dense quark matter, in the CFL phase and therefore a transparent insulator. We calculate the angles of reflection and refraction, and the intensity of the reflected and refracted light. Although the only potentially observable context for this phenomenon (reflection of light from and refraction of light through an illuminated quark star) is unlikely to be realized, our calculation casts new light on the old idea that confinement makes the QCD vacuum behave as if filled with a condensate of color-magnetic monopoles

Manuel, Cristina

Rajagopal, Krishna

CERN Document Server

Illuminating Dense Quark MatterCristina Manuela and Krishna RajagopalbaTheory Division, CERN, CH-1211 Geneva 23, Switzerland.bCenter for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USAWe imagine shining light on a lump of cold dense quark matter, in the CFL phase and therefore atransparent insulator. We calculate the angles of reflection and refraction, and the intensity of thereflected and refracted light. Although the only potentially observable context for this phenomenon(reflection of light from and refraction of light through an illuminated quark star) is unlikely to berealized, our calculation casts new light on the old idea that connement makes the QCD vacuumbehave as if lled with a condensate of color-magnetic monopoles.PACS numbers:At high enough baryon density and low temperature,the ground state of QCD with three flavors of quarksis the color-flavor locked (CFL) phase [1,2]. In thisphase, quarks of all three colors and all three flavors formCooper pairs, meaning that all fermionic quasiparticlesare gapped. The gap  is likely of order tens to 100 MeVat astrophysically accessible densities, with quark chemi-cal potential µ  (350−500) MeV [2]. The condensate ischarged with respect to eight of the nine massless gaugebosons (eight gluons, one photon) of the ordinary vac-uum, meaning that eight gauge bosons get a mass. Chi-ral symmetry is spontaneously broken, and so is baryonnumber (i.e., the material is a superfluid.) At asymptoticdensities, the eective coupling is weak and the proper-ties of the ground state and its low-energy excitationscan be determined quantitatively by adapting methodsused in the theory of superconductivity (BCS theory).The CFL phase persists for nite masses and evenfor unequal masses, so long as the dierences are nottoo large [3]. It is very likely the ground state for realQCD, assumed to be in equilibrium with respect to theweak interactions, over a substantial range of densities.Throughout the range of parameters over which the CFLphase exists as a bulk (and therefore electrically neu-tral) phase, it consists of equal numbers of u, d and squarks and is therefore electrically neutral in the absenceof any electrons [4]. The equality of the three quark num-ber densities is enforced in the CFL phase by the factthat this equality maximizes the pairing energy associ-ated with the formation of ud, us, and ds Cooper pairs.This equality is enforced even though the strange quark,with mass ms, is heavier than the light quarks [5].In the CFL phase, there is an unbroken U(1)Q˜ gaugesymmetry and a corresponding massless photon given bya linear combination of the ordinary photon and one ofthe gluons [1,6,2]. U(1)Q˜ is generated by ~Q = Q+T8/p3,where Q is the conventional electromagnetic charge gen-erator and the color hypercharge generator T8 is nor-malized such that, in the representation of the quarks,T8/p3 = diag(− 23, 13, 13) in color space. The CFL con-densate is ~Q-neutral, the U(1) symmetry generated by ~Qis therefore unbroken, the associated ~Q-photon remainsmassless, and within the CFL phase the ~Q-electric and~Q-magnetic elds satisfy Maxwell’s equations. The mass-less combination of the photon and the eighth gluon, AQ˜µ ,and the orthogonal massive combination which experi-ences the Meissner eect, AXµ , are given byAQ˜µ = cos θAµ + sin θG8µ , (1)AXµ = − sin θAµ + cos θG8µ . (2)The mixing angle θ (called α0 in Ref. [6]) which speciesthe unbroken U(1) is given bycos θ =g√g2 + e2/3. (3)θ is the analogue of the Weinberg angle in electroweaktheory. At accessible densities, the gluons are stronglycoupled (g2/4pi  1) and the photons are weakly cou-pled (e2/4pi  1/137), so θ is small, perhaps of order1/20. The \rotated photon" consists mostly of the usualphoton, with only a small admixture of the G8 gluon.All elementary excitations in the CFL phase are either~Q-neutral or couple to AQ˜µ with charges which are integermultiples of the ~Q-charge of the electron ~e = e cos θ,which is less than e because the electron couples only tothe Aµ component of AQ˜µ . The only massless excitation(the superfluid mode) is ~Q-neutral. Because all chargedexcitations have nonzero mass and there are no electronspresent, the CFL phase in bulk is a transparent insulatorat low temperatures: ~Q-magnetic and ~Q-electric eldswithin it evolve simply according to Maxwell’s equations,and low frequency ~Q-light traverses it without scattering.Imagine shining light on a chunk of dense quark mat-ter in the CFL phase. If CFL matter occurs only withinthe cores of neutron stars, cloaked under kilometers ofhadronic matter [7], the thought experiment we describehere in which light waves travelling in vacuum strike CFLmatter can never arise in nature. If, however, the factthat quark matter features many more strange quarksthan ordinary nuclear matter renders it stable even atzero pressure, then one may imagine quark stars in na-ture [8]. Such a quark star may be made of CFL quark1matter throughout, or may have an outer layer in whicha less symmetric pattern of pairing occurs. For example,quarks of only two colors and flavors may pair, yieldingthe 2SC phase which was the rst color superconductingphase studied [9]. Some of the remaining quarks with dif-fering Fermi momenta may also form a crystalline colorsuperconductor [10]. As in the CFL phase, the 2SC con-densate leaves a (slightly dierent) ~Q-photon massless.However, the 2SC phase is a good ~Q-conductor becauseof the presence of unpaired quarks and electrons. Thus,2SC matter is opaque and metallic rather than trans-parent and insulating. Illuminating it would result inabsorption and reflection, but no refraction. We shallassume that the quark matter we illuminate is in thetransparent CFL phase all the way out to its surface.Consider, then, an enormously dense, but transparent,illuminated quark star. Some light falling on its surfacewill reflect, and some will refract into the star in the formof ~Q-light. We shall calculate the reflection and refrac-tion angles and the intensity of the reflected light andrefracted ~Q-light. The partial Meissner eect induced bya static magnetic eld has been analyzed previously [6].We analyze a time-varying electromagnetic eld. As abonus, our analysis allows us to use well understood prop-erties of dense quark matter in the CFL phase to learnabout the (less well understood) QCD vacuum.We assume that the light has ω and k both much lessthan the energy needed to create a charged excitation inthe CFL phase. This means ω, k  , where  is thefermionic gap, to avoid the breaking of pairs and the cre-ation of quasiparticles. It also means ω, k  mpi± , mK± ,where pi and K are the charged pions and kaons of theCFL phase. Their masses are of order ~e in the chirallimit [11], and the contribution to their masses due tonite quark masses has also been evaluated [12].In vacuum the electromagnetic elds obey the freeMaxwell’s equationsr D = 0 , rE = −∂B∂t, (4a)r B = 0 , rH = ∂D∂t, (4b)where D = 0E and B = µ0H, and 0 and µ0 are thevacuum dielectric constant and magnetic permeability,respectively, such that the velocity of light c = 1/pµ00.Deep in the CFL phase, the rotated elds ~E and ~B obeythe same eld equations, but with dielectric constant [13]~ = 0(1 +8α9picos2 θµ22), (5)where α is the electromagnetic ne structure constantand µ is the chemical potential. This expression for ~is valid to leading order in α, and for ω, k  . Thedependence of ~ on ω arises only in corrections to (5)which are suppressed by ω2/2, and we therefore neglectdispersion in this letter. The magnetic permeability inthe CFL phase remains unchanged to leading order, ~µ =µ0. The index of refraction of CFL quark matter thusreduces to ~n =√~µ~/µ00 =√~/0. If we apply (5) forµ/  (4− 10), we obtain ~n  (1.02− 1.1).We take the surface of the CFL matter to be planar,with the CFL phase at z > 0 and vacuum at z < 0. (Thatis, we assume any curvature of the surface is on lengthscales long compared to the wavelength of the light.) Foran ordinary dielectric, the analogous problem is solved inRef. [14]. The complication here is that we must matchthe ordinary electric and magnetic elds in vacuum onto~Q-electric and ~Q-magnetic elds within the CFL phase.The properties of the reflected and refracted waves willtherefore depend upon both the dielectric constant ~ andthe mixing angle θ.We are only interested in the reflected and refractedwaves, and not in the detailed eld congurations veryclose to the interface. This means that we can followthe strategy of Ref. [6] and encapsulate the physics ofthe interface into boundary conditions relating E and Bon the vacuum side of the interface to ~E and ~B on theCFL side. On the CFL side, the massive X elds canbe neglected as long as z is greater than some λCFL,while on the vacuum side, the conned gluon elds canbe neglected as long as jzj is greater than some λQCD.λQCD is a length scale characteristic of connement. Forthe non-static elds of interest, and in the weak couplingregime, λCFL is of order 1/, longer than the inverseMeissner mass  gµ [15]. In order to describe light whosewavelength is long compared to λQCD and λCFL, we needboundary conditions relating E and B at z = −λQCD to~E and ~B at z = +λCFL.X-magnetic elds experience a Meissner eect in theCFL phase, meaning that supercurrents in the CFL mat-ter within λCFL of the interface screen the X-componentof any ordinary magnetic eld parallel to the interface onthe vacuum side, yielding the boundary condition~Hk(t, x, y, λCFL) = cos θ Hk(t, x, y,−λQCD) . (6)The CFL condensate is charged with respect to the Xgauge boson, meaning that if there is an ordinary electriceld perpendicular to the interface on the vacuum side,the X component of the electric flux will terminate inthe CFL phase within λCFL of the interface, yielding~D?(t, x, y, λCFL) = cos θ D?(t, x, y,−λQCD) . (7)We expect that the conned QCD vacuum should behaveas if it is a condensate of color-magnetic monopoles [16].That is, in the vacuum color magnetic eld lines end: ifthere is a ~Q-magnetic eld perpendicular to the interfaceon the CFL side, the vacuum will ensure that only theordinary magnetic eld is admitted. Thus,B?(t, x, y,−λQCD) = cos θ ~B?(t, x, y, λCFL) . (8)Finally, color magnetic currents on the vacuum side ofthe interface should exclude the color component of any2~Q-electric eld parallel to the interface on the CFL side,ensuring thatEk(t, x, y,−λQCD) = cos θ ~Ek(t, x, y, λCFL) . (9)At suciently high density, the property of CFL matterfrom which (6) and (7) follow, namely the Meissner eectfor X-bosons, is a weak-coupling phenomenon which canbe understood analytically. The properties of the QCDvacuum used to deduce (8) and (9) follow from a rea-sonable and familiar description of connement as a dualMeissner eect, but connement is not yet understoodanalytically. It is therefore of interest that our analysisbelow provides a derivation of (8) and (9) from (6).Consider an incident wave, with wave vector k =ωc (sin i, 0, cos i), so thatE = Eei(kr−ωt) , B = pµ00 kkE . (10)There are two orthogonal linear polarizations, shown inFig. 1, which we will treat separately. In the rst, thevector E is parallel to the interface while in the second,E lies in the plane of incidence. The reflected wave isE0 = E 0ei(k′r−ωt) , B0 = pµ00 k0k0E0 , (11)with wave vector k0 = ωc (sin i0, 0,− cos i0), while the re-fracted wave is~Er = ~Erei(krr−ωt) , ~Br =√~µ~krkr ~Er , (12)with wave vector kr = ωp~µ~(sin r, 0, cos r).The boundary conditions must be obeyed at all times,which immediately implies that the frequency of all thewaves is the same, as above. The boundary conditionsmust be obeyed at all points on the planar interface. For1/k  λCFL, λQCD this implies that k  r = k0  r =kr  r at z = 0, independent of details of the boundaryconditions. To satisfy this kinematic constraint, all threewave vectors must lie in a plane and k sin i = k0 sin i0 =kr sin r. Since k = k0, we must have i = i0: that is, theangle of incidence is the same as the angle of reflection.Since kr = ~nk, we also reproduce Snell’s lawsin i = ~n sin r . (13)The kinematics of the reflection and refraction of light onCFL quark matter are unaected by the mixing angle θ.We now use the boundary conditions to nd the in-tensities of the reflected and refracted radiation. For therst polarization in Fig. 1, (6) and (9) yieldcos θ (E − E 0)√0µ0cos i = ~Er√~~µcos r ,E + E 0 = cos θ ~Er , (14)and, using Snell’s law, (8) is equivalent to (9) in this case.Solving, we ndFIG. 1. Incident wave k strikes a planar interface be-tween vacuum and CFL quark matter, giving a reflected wavek′ and a refracted wave kr. We use boundary conditionsto relate electromagnetic elds just below the interface to~Q-electromagnetic elds just above, assuming that the wave-length of the light is long compared to the screening length inthe CFL phase and the connement length in vacuum, sym-bolized by grey shading. Top panel: polarization perpendicu-lar to the plane of incidence and thus parallel to the interface.Bottom panel: polarization parallel to the plane of incidence.~ErE =2 cos θ cos icos2 θ cos i + µ0µ˜ ~n cos r, (15a)E 0E =cos2 θ cos i− µ0µ˜ ~n cos rcos2 θ cos i + µ0µ˜ ~n cos r, (15b)where r is easily eliminated using Snell’s law in the form~n cos r =√~n2 − sin2 i. To the order we are working,~µ = µ0. For the second polarization of Fig. 1, (9) andeither (6) or (7) yield(E − E 0) cos i = cos θ ~Er cos r ,cos θ√0µ0(E + E 0) =√~~µ~Er , (16)3and hence~ErE =2~n cos θ cos iµ0µ˜ ~n2 cos i + ~n cos r cos2 θ, (17a)E 0E =µ0µ˜ ~n2 cos i− ~n cos r cos2 θµ0µ˜ ~n2 cos i + ~n cos r cos2 θ. (17b)Upon setting cos θ = 1, the relations (15) and (17) repro-duce results for reflection and refraction o standard di-electric media (see Ref. [14]). Decreasing cos θ decreasesthe Aµ component of AQ˜µ , and thus favors reflection overrefraction. For θ as small as in nature, the changes in-troduced by θ 6= 0 are small. In the (unphysical) limit inwhich cos θ = 0, AQ˜µ would be orthogonal to Aµ makingthe CFL phase a superconductor with respect to ordinaryelectromagnetism. In this limit, we expect and nd zerorefraction and perfect reflection for both polarizations.The value of Brewster’s angle is modied by a non-vanishing θ. This is the incident angle for which thereis no reflected wave if the polarization is parallel to theplane of incidence. We ndiB = arctan(~ncos2 θ√~n2 − cos4 θ~n2 − 1). (18)We can also imagine sending a ~Q electromagnetic wavefrom CFL matter into vacuum. Doing this calculationyields the same results, but with ~n replaced by 1/~n. Asusual, Snell’s law then implies that for incident anglesbigger than arcsin (1/~n), the ~Q-light cannot escape fromthe CFL matter.Are the solutions (15) and (17) consistent with energyconservation? The Poynting vector S = 12 (EH) mea-sures the energy flow per unit area and time. Continuityof the z-component of the Poynting vector requiresE2√0µ0cos i = (E 0)2√0µ0cos i + ( ~Er)2√~~µcos r , (19)a relation which is indeed satised by both (15) and (17).Notice that in our analysis of each of the two polariza-tions, one boundary condition was irrelevant and Snell’slaw could be used to eliminate a second. If we had usedenergy conservation in the form (19) in our derivationinstead of just as a check, we could have derived all ourresults from the single boundary condition (6). That is,given only the boundary condition (6) which is easily de-rived, Snell’s law (13) which is kinematic, and energyconservation (19), we can derive our solutions describingthe reflection and refraction of light of both polarizationsand, from these electromagnetic elds, we can then de-rive the remaining boundary conditions (7), (8) and (9).This means that we have derived the boundary condi-tions motivated above by the idea that the QCD vacuumbehaves like a dual superconductor lled with a conden-sate of color-magnetic monopoles [16]. Having analyzedthe illumination of dense quark matter, we nd that inaddition we have illuminated our understanding of theQCD vacuum.We thank D. Son for a stimulating conversation. CMthanks the MIT CTP for hospitality during an early stageof this work. The work of CM is supported by the Euro-pean Community through Marie Curie Grant #HPMF-CT-1999-00391. The work of KR is supported in part bythe DOE under agreement #DF-FC02-94ER40818 andan OJI grant, and by the A. P. Sloan Foundation.[1] M. Alford, K. Rajagopal and F. Wilczek, Nucl. Phys.B537, 443 (1999).[2] For reviews, see K. Rajagopal and F. Wilczek, hep-ph/0011333; M. Alford, hep-ph/0102047.[3] M. Alford, J. Berges and K. Rajagopal, Nucl. Phys.B558, 219 (1999); T. Scha¨fer and F. Wilczek, Phys. Rev.D60, 074014 (1999).[4] K. Rajagopal and F. Wilczek, Phys. Rev. Lett. 86, 3492(2001).[5] If higher order eects introduce a K0 condensate, as sug-gested in P. F. Bedaque and T. Scha¨fer, hep-ph/0105150,the conclusion that the CFL phase is electrically neutralin the absence of electrons is unchanged, although theargument for this conclusion becomes more involved.[6] M. Alford, J. Berges and K. Rajagopal, Nucl. Phys.B571, 269 (2000).[7] M. Alford, K. Rajagopal, S. Reddy and F. Wilczek, hep-ph/0105009.[8] E. Witten, Phys. Rev. D30, 272 (1984); E. Farhi andR. L. Jae, Phys. Rev. D30, 2379 (1984); P. Haensel, J.L. Zdunik and R. Schaeer, Astron. Astrophys. 160, 121(1986); C. Alcock, E. Farhi and A. Olinto, Phys. Rev.Lett. 57, 2088 (1986); Astrophys. J. 310, 261 (1986).[9] B. Barrois, Nucl. Phys. B129, 390 (1977); D. Bailin andA. Love, Phys. Rept. 107, 325 (1984); M. Alford, K.Rajagopal and F. Wilczek, Phys. Lett. B422, 247 (1998);R. Rapp, T. Scha¨fer, E. V. Shuryak and M. Velkovsky,Phys. Rev. Lett. 81, 53 (1998).[10] M. Alford, J. Bowers and K. Rajagopal, Phys. Rev. D63, 074016 (2001); A. Leibovich, K. Rajagopal and E.Shuster, hep-ph/0104073.[11] C. Manuel and M. H. Tytgat, Phys. Lett. B 501, 200(2001).[12] D. T. Son and M. A. Stephanov, Phys. Rev. D61, 074012(2000); erratum, ibid. D62, 059902 (2000).[13] D. F. Litim and C. Manuel, hep-ph/0105165.[14] J. D. Jackson, \Classical Electrodynamics", 3rd Ed.,(Wiley, New York, 1998).[15] D. H. Rischke, Phys. Rev. D 62, 054017 (2000); R. Casal-buoni, R. Gatto and G. Nardulli, Phys. Lett. B 498, 179(2001).[16] G. ’t Hooft, in \High Energy Physics," ed. A. Zichichi(Editrice Compositori, Bologna, 1976); S. Mandelstam,Phys. Rept. 23, 245 (1976).4

Illuminating Dense Quark Matter

Abstract

Similar works

Full text

Available Versions

Crossref

CERN Document Server